Physics 8 Monday, September 9, 2013

Transcription

1 Physics 8 Monday, September 9, 2013 HW2 (due Friday) printed copies. Read Chapter 4 (momentum) for Wednesday. I m reading through the book with you. It s my 3rd time now. One purpose of the reading responses is for you to convince me that you really read the chapter. Please answer with enough detail to make this obvious to me. (Most of you do.) Bill hosts HW study/help sessions on Thursdays at 7pm in DRL 3W2. Zoey hosts HW sessions on Wednesdays at 7pm in DRL 2N36. Free physics tutoring (by physics majors) for intro physics courses, M Th 3 7pm in Room 253 of Education Commons in Weiss Pavillion.

2 Modified version of HW1 #7. A husband and wife work in buildings ten blocks apart and plan to meet for lunch. The husband strolls at 1.0 m/s, while the wife walks briskly at 1.5 m/s. Knowing this, the wife picks a restaurant between the two buildings at which she and her husband will arrive at the same instant, if the two leave their respective buildings at the same instant. In blocks, how far from the wife s building is the restaurant? answer: 6 blocks

3 Examples of excellent responses to Chapter Acceleration is the rate of change of velocity. So any time an objects velocity is changing (ie. it is not traveling at a constant rate) it has some sort of acceleration (which can be positive or negative). Average acceleration is the change in overall velocity (final minus initial) divided by the time. The relationship between acceleration and velocity is very similar to that of velocity and displacement. 2. The steel ball becomes a projectile once it is dropped, because the only force acting on it is gravity. It is dropped from rest, so its initial velocity is zero, but the ball has an increasing speed as it falls. This is because of its constant acceleration of g (9.8 m/s 2 ) due to gravity). Its velocity will increase at this rate until it reaches the ground, where its velocity and acceleration will go to zero.

4 3. There wasn t anything particularly difficult to understand. But I learned that in projectile motion in which an object is thrown vertically upward, the acceleration of the object is never zero at any point of its trajectory, even at the very top. The object needs to change its velocity when it gets to the top of the trajectory and it needs acceleration to change its velocity. Therefore, [acceleration is downward] even when it s at the topmost part [where it has zero velocity]. This seemed interesting to me. 3. Since I took Physics in high school, a lot of this was mainly review. Nonetheless, when I last took Physics I had not yet taken Calculus or understood what a derivative is. For this reason, I still find it difficult conceptually to relate velocity and acceleration, in terms of derivatives. In addition, the motion diagram and kinematics graphs for the three basic types of motion were particularly hard to envision (even though they were presented visually).

5 Common responses to today s reading Waking up one s Calculus brain. Instantaneous acceleration. Galileo: x t 2. Trigonometry & inclined planes. Motion diagrams. Getting through all the math at the end of the chapter. Interesting: positive acceleration (a x > 0) for an object with negative velocity (v x < 0) actually slows down the object!

6 Key results from Chapter 3: defining acceleration Last week, we defined velocity v x = dx dt (considering only the x component for now), and we learned to identify v x visually as the slope on a graph of x(t) Moving at constant velocity is not very interesting! So we need to be able to talk about changes in velocity. The rate of change of velocity with time is called acceleration: a x = dv x dt While acceleration can also vary with time (!), there are many situations in which constant acceleration (a x = constant) gives a good description of the motion.

7 At time t 2 in the position-vs-time graph below, the object is (a) not moving (b) moving at constant speed (c) speeding up (d) slowing down

8 At time t 2 in the position-vs-time graph below, is v x (the x component of velocity) (a) zero (b) not changing (c) increasing (d) decreasing

9 The x component of acceleration in these two graphs is (A) positive in (a), negative in (b) (B) negative in (a), positive in (b) (C) negative in both (a) and (b) (D) positive in both (a) and (b) (E) zero in both (a) and (b)

10 Accelerating under gravity s influence One important situation in which constant acceleration (a x = constant) gives a good description of the motion is free fall near Earth s surface. Free fall is the motion of an object subject only to the influence of gravity. Not being pushed or held by your hand or by the ground When air resistance is small enough to neglect Close to Earth s surface, an object in free fall experiences a constant acceleration, of magnitude a = 9.8 m/s 2 and pointing in the downward direction. If we define the x axis to point upward (which we often do, but you need to check this), then a x = 9.8 m/s 2. Since we see the quantity 9.8 m/s 2 so often, we give it a name: g = 9.8 m/s 2. Then a x = g.

11 (Checkpoint 3.7) Let s pause here to go through Checkpoint 3.7 together. Does the speed of a falling object (a) increase or (b) decrease? If the positive x axis points up, does v x (a) increase or (b) decrease as the object falls? is the x component of the acceleration (a) positive or (b) negative? Argue with your neighbor for a moment, and then we ll shout out answers together.

12 Let s do what Galileo could only imagine doing! Let s see if different objects really do fall with the same acceleration a x = g if we are able to remove the effects of air resistance.

13 Equations we can derive from a x = constant We defined a x = dvx dt and v x = dx dt, without worrying so far about whether or not a x is changing with time. Integrating the first equation over time, t v x (t) = v x (0) + a x dt 0 If a x = constant, then this integral becomes easy: v x (t) = v x (0) + a x t We can also try to integrate the second equation over time: t x(t) = x(0) + v x dt 0 keeping in mind that v x (unlike a x ) is changing with time

14 Equations we can derive from a x = constant Plugging our v x (t) result into the second integal: t x(t) = x(0) + (v x (0) + a x t) dt 0 x(t) = x(0) + v x (0)t a xt 2 Using subscript i to mean initial for t = 0 values: x(t) = x i + v x,i t a xt 2

15 Equations we can derive from a x = constant That s all there is to it. Simply writing down the assumption that a x is constant allows us to integrate twice to get v x,f = v x,i + a x t x f = x i + v x,i t a xt 2 If you plug one of these equations into the other, you can eliminate t to get one more useful result v 2 x,f = v 2 x,i + 2a x (x f x i ) This last one is helpful e.g. to know how fast the dropped steel ball is traveling at the instant before it hits the ground. My point is that these equations are just the result of taking a x = constant and doing some math.

16 Inclined planes Falling to the ground at a x = g happens so quickly that it can be difficult to see exactly what is happening. Maybe there is a way to fall in slow motion? Yes! We can slide down a hill. g g sin θ (We ll see in Chapter 10 why it s sin θ here. Don t worry.) To get the ± sign right, you have to choose which direction to draw x axis. Textbook chooses x axis to point downhill a x = +g sin θ Let s look at this contraption and figure out which way it defines the x axis to point

17 Inclined air track It looks as if the x axis points downhill, and the point on the top of the ramp is called x = 0.

18 Which of the following shows the expected shapes of x(t) [blue] and v x (t) [red] if I release the cart (at rest) from x = 0? C D A B

19 Which of the following shows the expected shapes of x(t) [blue] and v x (t) [red] if I shove the cart upward starting from x = +2 m? C D A B

20 I shove the cart uphill and watch it travel up and back. We define the x axis to point downhill. At the top of its trajectory (where it turns around), v x is (a) positive (b) negative (c) zero (d) infinite (e) undefined

21 I shove the cart uphill and watch it travel up and back. We define the x axis to point downhill. At the top of its trajectory (where it turns around), a x is (a) positive (b) negative (c) zero (d) infinite (e) undefined

22 Physics 8 Monday, September 9, 2013 HW2 (due Friday) printed copies. Read Chapter 4 (momentum) for Wednesday. I m reading through the book with you. It s my 3rd time now. One purpose of the reading responses is for you to convince me that you really read the chapter. Please answer with enough detail to make this obvious to me. (Most of you do.) Bill hosts HW study/help sessions on Thursdays at 7pm in DRL 3W2. Zoey hosts HW sessions on Wednesdays at 7pm in DRL 2N36. Free physics tutoring (by physics majors) for intro physics courses, M Th 3 7pm in Room 253 of Education Commons in Weiss Pavillion. I ll put today s slides up on Canvas this afternoon.